A second-order optimality condition with first- and second-order complementarity associated with global convergence of algorithms

Gabriel Haeser ()
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Gabriel Haeser: University of São Paulo

Computational Optimization and Applications, 2018, vol. 70, issue 2, No 11, 615-639

Abstract: Abstract We develop a new notion of second-order complementarity with respect to the tangent subspace related to second-order necessary optimality conditions by the introduction of so-called tangent multipliers. We prove that around a local minimizer, a second-order stationarity residual can be driven to zero while controlling the growth of Lagrange multipliers and tangent multipliers, which gives a new second-order optimality condition without constraint qualifications stronger than previous ones associated with global convergence of algorithms. We prove that second-order variants of augmented Lagrangian (under an additional smoothness assumption based on the Lojasiewicz inequality) and interior point methods generate sequences satisfying our optimality condition. We present also a companion minimal constraint qualification, weaker than the ones known for second-order methods, that ensures usual global convergence results to a classical second-order stationary point. Finally, our optimality condition naturally suggests a definition of second-order stationarity suitable for the computation of iteration complexity bounds and for the definition of stopping criteria.

Keywords: Second-order optimality conditions; Complementarity; Global convergence; Constraint qualifications; 90C46; 90C30 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10589-018-0005-3

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